The method of maximum likelihood corresponds to many well-known estimation methods in statistics. For example, one may be interested in the heights of adult female penguins, but be unable to measure the height of every single penguin in a population due to cost or time constraints. Assuming that the heights are normally (Gaussian) distributed with some unknown mean and variance, the mean and variance can be estimated with MLE while only knowing the heights of some sample of the overall population. MLE would accomplish this by taking the mean and variance as parameters and finding particular parametric values that make the observed results the most probable (given the model).

In general, for a fixed set of data and underlying statistical model, the method of maximum likelihood selects the set of values of the model parameters that maximizes the likelihood function. Intuitively, this maximizes the "agreement" of the selected model with the observed data, and for discrete random variables it indeed maximizes the probability of the observed data under the resulting distribution. Maximum-likelihood estimation gives a unified approach to estimation, which is well-defined in the case of the normal distribution and many other problems. However, in some complicated problems, difficulties do occur: in such problems, maximum-likelihood estimators are unsuitable or do not exist.[citation needed]

Suppose there is a sample x1, x2, …, xn of nindependent and identically distributed observations, coming from a distribution with an unknown probability density functionf0(·). It is however surmised that the function f0 belongs to a certain family of distributions { f(·| θ), θ ∈ Θ } (where θ is a vector of parameters for this family), called the parametric model, so that f0 = f(·| θ0). The value θ0 is unknown and is referred to as the true value of the parameter vector. It is desirable to find an estimator which would be as close to the true value θ0 as possible. Either or both the observed variables xi and the parameter θ can be vectors.

Now we look at this function from a different perspective by considering the observed values x1, x2, …, xn to be fixed "parameters" of this function, whereas θ will be the function's variable and allowed to vary freely; this function will be called the likelihood:

Note denotes a separation between the two input arguments: and the vector-valued input .

In practice it is often more convenient to work with the logarithm of the likelihood function, called the log-likelihood:

or the average log-likelihood:

The hat over ℓ indicates that it is akin to some estimator. Indeed, estimates the expected log-likelihood of a single observation in the model.

The method of maximum likelihood estimates θ0 by finding a value of θ that maximizes . This method of estimation defines a maximum-likelihood estimator (MLE) of θ0 …

… if any maximum exists. An MLE estimate is the same regardless of whether we maximize the likelihood or the log-likelihood function, since log is a strictly monotonically increasing function.

For many models, a maximum likelihood estimator can be found as an explicit function of the observed data x1, …, xn. For many other models, however, no closed-form solution to the maximization problem is known or available, and an MLE has to be found numerically using optimizationmethods. For some problems, there may be multiple estimates that maximize the likelihood. For other problems, no maximum likelihood estimate exists (meaning that the log-likelihood function increases without attaining the supremum value).

In the exposition above, it is assumed that the data are independent and identically distributed. The method can be applied however to a broader setting, as long as it is possible to write the joint density function f(x1, …, xn | θ), and its parameter θ has a finite dimension which does not depend on the sample size n. In a simpler extension, an allowance can be made for data heterogeneity, so that the joint density is equal to f1(x1|θ) · f2(x2|θ) · ··· · fn(xn | θ). Put another way, we are now assuming that each observation xi comes from a random variable that has its own distribution function fi. In the more complicated case of time series models, the independence assumption may have to be dropped as well.

where is the prior distribution for the parameter θ and where is the probability of the data averaged over all parameters. Since the denominator is independent of θ, the Bayesian estimator is obtained by maximizing with respect to θ. If we further assume that the prior is a uniform distribution, the Bayesian estimator is obtained by maximizing the likelihood function . Thus the Bayesian estimator coincides with the maximum-likelihood estimator for a uniform prior distribution .

this being the sample analogue of the expected log-likelihood , where this expectation is taken with respect to the true density .

Maximum-likelihood estimators have no optimum properties for finite samples, in the sense that (when evaluated on finite samples) other estimators may have greater concentration around the true parameter-value.[1] However, like other estimation methods, maximum-likelihood estimation possesses a number of attractive limiting properties: As the sample size increases to infinity, sequences of maximum-likelihood estimators have these properties:

Consistency: the sequence of MLEs converges in probability to the value being estimated.

Asymptotic normality: as the sample size increases, the distribution of the MLE tends to the Gaussian distribution with mean and covariance matrix equal to the inverse of the Fisher information matrix.

Under the conditions outlined below, the maximum likelihood estimator is consistent. The consistency means that having a sufficiently large number of observations n, it is possible to find the value of θ0 with arbitrary precision. In mathematical terms this means that as n goes to infinity the estimator converges in probability to its true value:

In other words, different parameter values θ correspond to different distributions within the model. If this condition did not hold, there would be some value θ1 such that θ0 and θ1 generate an identical distribution of the observable data. Then we wouldn't be able to distinguish between these two parameters even with an infinite amount of data — these parameters would have been observationally equivalent.

The identification condition is absolutely necessary for the ML estimator to be consistent. When this condition holds, the limiting likelihood function ℓ(θ|·) has unique global maximum at θ0.

The identification condition establishes that the log-likelihood has a unique global maximum. Compactness implies that the likelihood cannot approach the maximum value arbitrarily close at some other point (as demonstrated for example in the picture on the right).

Compactness is only a sufficient condition and not a necessary condition. Compactness can be replaced by some other conditions, such as:

both concavity of the log-likelihood function and compactness of some (nonempty) upper level sets of the log-likelihood function, or

existence of a compact neighborhood N of θ0 such that outside of N the log-likelihood function is less than the maximum by at least some ε > 0.

Continuity: the function ln f(x|θ) is continuous in θ for almost all values of x:

Dominance: there exists D(x) integrable with respect to the distribution f(x|θ0) such that

By the uniform law of large numbers, the dominance condition together with continuity establish the uniform convergence in probability of the log-likelihood:

The dominance condition can be employed in the case of i.i.d. observations. In the non-i.i.d. case the uniform convergence in probability can be checked by showing that the sequence is stochastically equicontinuous.

If one wants to demonstrate that the ML estimator converges to θ0almost surely, then a stronger condition of uniform convergence almost surely has to be imposed:

Maximum-likelihood estimators can lack asymptotic normality and can be inconsistent if there is a failure of one (or more) of the below regularity conditions:

Estimate on boundary. Sometimes the maximum likelihood estimate lies on the boundary of the set of possible parameters, or (if the boundary is not, strictly speaking, allowed) the likelihood gets larger and larger as the parameter approaches the boundary. Standard asymptotic theory needs the assumption that the true parameter value lies away from the boundary. If we have enough data, the maximum likelihood estimate will keep away from the boundary too. But with smaller samples, the estimate can lie on the boundary. In such cases, the asymptotic theory clearly does not give a practically useful approximation. Examples here would be variance-component models, where each component of variance, σ2, must satisfy the constraint σ2 ≥0.

Data boundary parameter-dependent. For the theory to apply in a simple way, the set of data values which has positive probability (or positive probability density) should not depend on the unknown parameter. A simple example where such parameter-dependence does hold is the case of estimating θ from a set of independent identically distributed when the common distribution is uniform on the range (0,θ). For estimation purposes the relevant range of θ is such that θ cannot be less than the largest observation. Because the interval (0,θ) is not compact, there exists no maximum for the likelihood function: For any estimate of theta, there exists a greater estimate that also has greater likelihood. In contrast, the interval [0,θ] includes the end-point θ and is compact, in which case the maximum-likelihood estimator exists. However, in this case, the maximum-likelihood estimator is biased. Asymptotically, this maximum-likelihood estimator is not normally distributed.[3]

Nuisance parameters. For maximum likelihood estimations, a model may have a number of nuisance parameters. For the asymptotic behaviour outlined to hold, the number of nuisance parameters should not increase with the number of observations (the sample size). A well-known example of this case is where observations occur as pairs, where the observations in each pair have a different (unknown) mean but otherwise the observations are independent and normally distributed with a common variance. Here for 2N observations, there are N+1 parameters. It is well known that the maximum likelihood estimate for the variance does not converge to the true value of the variance.

Increasing information. For the asymptotics to hold in cases where the assumption of independent identically distributed observations does not hold, a basic requirement is that the amount of information in the data increases indefinitely as the sample size increases. Such a requirement may not be met if either there is too much dependence in the data (for example, if new observations are essentially identical to existing observations), or if new independent observations are subject to an increasing observation error.

Some regularity conditions which ensure this behavior are:

The first and second derivatives of the log-likelihood function must be defined.

The Fisher information matrix must not be zero, and must be continuous as a function of the parameter.

The maximum likelihood estimator selects the parameter value which gives the observed data the largest possible probability (or probability density, in the continuous case). If the parameter consists of a number of components, then we define their separate maximum likelihood estimators, as the corresponding component of the MLE of the complete parameter. Consistent with this, if is the MLE for θ, and if g(θ) is any transformation of θ, then the MLE for α = g(θ) is by definition

It maximizes the so-called profile likelihood:

The MLE is also invariant with respect to certain transformations of the data. If Y = g(X) where g is one to one and does not depend on the parameters to be estimated, then the density functions satisfy

and hence the likelihood functions for X and Y differ only by a factor that does not depend on the model parameters.

For example, the MLE parameters of the log-normal distribution are the same as those of the normal distribution fitted to the logarithm of the data.

In particular, it means that the bias of the maximum-likelihood estimator is equal to zero up to the order n−1/2. However when we consider the higher-order terms in the expansion of the distribution of this estimator, it turns out that θmle has bias of order n−1. This bias is equal to (componentwise)[5]

where Einstein's summation convention over the repeating indices has been adopted; Ijk denotes the j,k-th component of the inverse Fisher information matrix I−1, and

Using these formulas it is possible to estimate the second-order bias of the maximum likelihood estimator, and correct for that bias by subtracting it:

This estimator is unbiased up to the terms of order n−1, and is called the bias-corrected maximum likelihood estimator.

This bias-corrected estimator is second-order efficient (at least within the curved exponential family), meaning that it has minimal mean squared error among all second-order bias-corrected estimators, up to the terms of the order n−2. It is possible to continue this process, that is to derive the third-order bias-correction term, and so on. However as was shown by Kano (1996), the maximum-likelihood estimator is not third-order efficient.

Consider a case where n tickets numbered from 1 to n are placed in a box and one is selected at random (see uniform distribution); thus, the sample size is 1. If n is unknown, then the maximum-likelihood estimator of n is the number m on the drawn ticket. (The likelihood is 0 for n < m, 1/n for n ≥ m, and this is greatest when n = m. Note that the maximum likelihood estimate of n occurs at the lower extreme of possible values {m, m + 1, ...}, rather than somewhere in the "middle" of the range of possible values, which would result in less bias.) The expected value of the number m on the drawn ticket, and therefore the expected value of , is (n + 1)/2. As a result, with a sample size of 1, the maximum likelihood estimator for n will systematically underestimate n by (n − 1)/2.

Suppose one wishes to determine just how biased an unfair coin is. Call the probability of tossing a HEAD p. The goal then becomes to determine p.

Suppose the coin is tossed 80 times: i.e., the sample might be something like x1 = H, x2 = T, …, x80 = T, and the count of the number of HEADS "H" is observed.

The probability of tossing TAILS is 1 − p (so here p is θ above). Suppose the outcome is 49 HEADS and 31 TAILS, and suppose the coin was taken from a box containing three coins: one which gives HEADS with probability p = 1/3, one which gives HEADS with probability p = 1/2 and another which gives HEADS with probability p = 2/3. The coins have lost their labels, so which one it was is unknown. Using maximum likelihood estimation the coin that has the largest likelihood can be found, given the data that were observed. By using the probability mass function of the binomial distribution with sample size equal to 80, number successes equal to 49 but different values of p (the "probability of success"), the likelihood function (defined below) takes one of three values:

The likelihood is maximized when p = 2/3, and so this is the maximum likelihood estimate for p.

One way to maximize this function is by differentiating with respect to p and setting to zero:

which has solutions p = 0, p = 1, and p = 49/80. The solution which maximizes the likelihood is clearly p = 49/80 (since p = 0 and p = 1 result in a likelihood of zero). Thus the maximum likelihood estimator for p is 49/80.

This result is easily generalized by substituting a letter such as t in the place of 49 to represent the observed number of 'successes' of our Bernoulli trials, and a letter such as n in the place of 80 to represent the number of Bernoulli trials. Exactly the same calculation yields the maximum likelihood estimatort / n for any sequence of n Bernoulli trials resulting in t 'successes'.

This family of distributions has two parameters: θ = (μ, σ), so we maximize the likelihood, , over both parameters simultaneously, or if possible, individually.

Since the logarithm is a continuousstrictly increasing function over the range of the likelihood, the values which maximize the likelihood will also maximize its logarithm. This log likelihood can be written as follows:

This is indeed the maximum of the function since it is the only turning point in μ and the second derivative is strictly less than zero. Its expectation value is equal to the parameter μ of the given distribution,

which means that the maximum-likelihood estimator is unbiased.

Similarly we differentiate the log likelihood with respect to σ and equate to zero:

which is solved by

Inserting the estimate we obtain

To calculate its expected value, it is convenient to rewrite the expression in terms of zero-mean random variables (statistical error) . Expressing the estimate in these variables yields

Simplifying the expression above, utilizing the facts that and , allows us to obtain

This means that the estimator is biased. However, is consistent.

Formally we say that the maximum likelihood estimator for is:

In this case the MLEs could be obtained individually. In general this may not be the case, and the MLEs would have to be obtained simultaneously.

The normal log likelihood at its maximum takes a particularly simple form:

It may be the case that variables are correlated, that is, not independent. Two random variables X and Y are independent only if their joint probability density function is the product of the individual probability density functions, i.e.

Suppose one constructs an order-n Gaussian vector out of random variables , where each variable has means given by . Furthermore, let the covariance matrix be denoted by .

The joint probability density function of these n random variables is then given by:

In the two variable case, the joint probability density function is given by:

In this and other cases where a joint density function exists, the likelihood function is defined as above, in the section Principles, using this density.

Consider problems where both states and parameters such as require to be estimated. Iterative procedures such as Expectation-maximization algorithms may be used to solve joint state-parameter estimation problems.

For example, suppose that n samples of state estimates together with a sample mean have been calculated by either a minimum-variance Kalman filter or a minimum-variance smoother using a previous variance estimate . Then the next variance iterate may be obtained from the maximum likelihood estimate calculation

The convergence of MLEs within filtering and smoothing EM algorithms are studied in [6][7] .[8]

Mean squared error, a measure of how 'good' an estimator of a distributional parameter is (be it the maximum likelihood estimator or some other estimator).

The Rao–Blackwell theorem, a result which yields a process for finding the best possible unbiased estimator (in the sense of having minimal mean squared error). The MLE is often a good starting place for the process.

Sufficient statistic, a function of the data through which the MLE (if it exists and is unique) will depend on the data.